yeah, this sounds good. the points in the manifold are the points, and there may be many different coordinate systems that you can use to label the point, but we think of the point existing in some sense independently of which coordinate system we use.
you need n coordinates, where n is the dimension of the manifold.
only if the manifold is 1 dimensional (a curve)
if you label the elements of the set A with an index i, then you would use two indices (i,j) to label the elements of the set AxA. in the same sense, you need one real number to specify a point on the line (=R), 2 real numbers to specify a point on the plane (=RxR), etc.
i guess in a set theoretic sense, you don t actually need
two indices. the cartesian product of two sets has the same cardinality, so you could count the elements of the new set with the same index, if you had a mind to, but this would muck up the other nice properties of the set, so it is much preferred to just stick to the double index concept.
yes, its a good analogy
perhaps it would be useful to know the definition of a manifold. leaving out some technical details, the definition is this: an n-dimensional manifold is a space X such that for any point x in X, there is a neigborhood of x U and a continous invertible map with continuous inverse (homeomorphism) between U and Rn
in other words, if you look at a manifold up closely enough, it looks like Rn
. these local homeomorphisms are the charts of the manifold, they are the coordinates.
perhaps it would be useful to see an example. a sphere can be written as the map [itex](u,v)\mapsto(\sin u\cos v,\sin u\sin v,cos v)[/itex]. here u and v are the coordinates. there are two of them because a sphere is a 2 dimensional manifold
thats right. nothing at all (unless you like chroots approximations above. i don t.)
i m not sure what exactly you re saying here. i can assure you that, for example, if you have a smooth curve on the manifold, then this curve will have a tangent vector which is in the tangent space to the manifold at each point the curve passes through. furthermore, this tangent vector to the curve will vary from tangent space to tangent space, as the curve goes along, in a smooth way.
there is no discontinuity. perhaps it would be useful to know the following fact: a smooth manifold, together with the tangent space at each point, considered as one larger set, is also a smooth manifold. this manifold is called the tangent bundle.